The Stacks project

Lemma 110.7.1. There exists a local ring $R$ and a maximal ideal $\mathfrak m$ such that the completion $R^\wedge $ of $R$ with respect to $\mathfrak m$ has the following properties

  1. $R^\wedge $ is local, but its maximal ideal is not equal to $\mathfrak m R^\wedge $,

  2. $R^\wedge $ is not a complete local ring, and

  3. $R^\wedge $ is not $\mathfrak m$-adically complete as an $R$-module.

Proof. This follows from the discussion above as (with $R = k[x_1, x_2, x_3, \ldots ]$) the completion of the localization $R_{\mathfrak m}$ is equal to the completion of $R$. $\square$

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